Show that the averaged neutrino probabilities are given by...

Show that the averaged neutrino probabilities are given by Eq. (11.12) in a 2-flavor model described by Eq. (11.9). If for neutrinos in vacuum $\theta=20^{\circ}, E=5 \mathrm{MeV}$, and $\Delta m^2=10^{-5} \mathrm{eV}^2$, what is the minimum distance over which the probabilities must be averaged for Eq. (11.12) to be valid, and what is the averaged probability to detect an electron neutrino if the neutrino was produced as an electron neutrino?***

Key Concepts

Mixing Angle in Neutrino Oscillations

The mixing angle (often denoted as ?) quantifies the degree of mixing between the two neutrino flavors. It determines the amplitude of the oscillation probability; specifically, the term sin²2? in the oscillation probability governs how large the oscillation effects are. When the probability is averaged over many oscillations, the survival probability of a neutrino produced in one flavor is expressed in terms of this mixing angle.

Two-Flavor Neutrino Oscillations

This concept deals with describing neutrino flavor change in a model where only two neutrino flavors mix. In such models, the flavor states are linear combinations of the mass eigenstates, which leads to oscillations in the probability of detecting a specific neutrino flavor as it propagates. The formalism results in an oscillation probability that depends on the mixing angle, the neutrino energy, the difference in the squares of the masses of the neutrinos (?m²), and the distance traveled.

Oscillation Length

The oscillation length is the characteristic distance over which one complete cycle of the oscillation occurs. It is typically defined by L_osc = 4?E/?m² (in natural units). This length scale is crucial because if the neutrino’s travel distance exceeds the oscillation length, the probability oscillates rapidly, which is a key consideration when averaging the oscillatory behavior over space.

Probability Averaging

In many practical situations, especially in experiments where the neutrino source or detector cannot resolve rapid oscillations, the observed oscillation probability must be averaged over a distance that spans many cycles of the oscillation. When averaged, the sine-squared (oscillatory) term in the transition probability becomes effectively constant, yielding an averaged probability expression that is independent of distance. This approach leads to a simpler, averaged formula for the detection probability.

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